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The partial sums of a sequence ${\mathbf x} = x_1, x_2, \ldots, x_k$ ofdistinct non-identity elements of a group $(G,\cdot)$ are $s_0 = id_G$ and $s_j= \prod_{i=1}^j x_i$ for $0 < j \leq k$. If the partial sums are all differentthen ${\mathbf x}$ is a linear sequencing and if the partial sums are alldifferent when $|i-j| \leq t$ then ${\mathbf x}$ is a $t$-weak sequencing. Weinvestigate these notions of sequenceability in semidirect products using thepolynomial method. We show that every subset of order $k$ of the non-identityelements of the dihedral group of order $2m$ has a linear sequencing when $k\leq 12$ and either $m>3$ is prime or every prime factor of $m$ is larger than$k!$, unless $s_k$ is unavoidably the identity; that every subset of order $k$of a non-abelian group of order three times a prime has a linear sequencingwhen $5 < k \leq 10$, unless $s_k$ is unavoidably the identity; and that if theorder of a group is $pe$ then all sufficiently large subsets of thenon-identity elements are $t$-weakly sequenceable when $p>3$ is prime, $e \leq3$ and $t \leq 6$.

Sequencings in Semidirect Products via the Polynomial Method